It is obvious that \(\widehat{V}(|\psi _{AB}\rangle )\)  is a real
symmetric concave function of the Schmidt vector
\((\lambda _0,\lambda _1, \cdots , \lambda _{d-1})^\) . Therefore,
\(\widehat{V}(|\psi _{AB}\rangle )\)  is an entanglement measure for pure
state \(|\psi _{AB}\rangle \)  [1]}. Then the convex roof extension of the
standard symmetrized variance
\(\widehat{V}_c(\rho _{AB})=\min _{\lbrace p_k, |\psi ^{(k)}_{AB}\rangle \rbrace }\sum _k p_k \widehat{V}(|\psi ^{(k)}_{AB}\rangle )\) , where the minimum
runs over all possible pure state decompositions of
\(\rho _{AB}=\sum _k p_k |\psi ^{(k)}_{AB}\rangle \langle \psi ^{(k)}_{AB}|\) , is an entanglement measure for bipartite quantum
states. Recall that the well-known entanglement measure concurrence
of a pure state \(|\psi _{AB}\rangle \)  is defined by
\(\mathrm {E}(|\psi _{AB}\rangle )=\sqrt{2(1-\operatorname{Tr}_{}\left(\rho _A^2\right))}\)  with
\(\rho _A=Tr_B (|\psi _{AB}\rangle \langle \psi _{AB}|)\) , and
\(E(\rho _{AB})=\min _{\lbrace p_k, |\psi ^{(k)}_{AB}\rangle \rbrace } \sum _k p_k\mathrm {E}(|\psi _{AB}^{(k)}\rangle )\)  with the minimum running over all
possible pure state decompositions of \(\rho _{AB}=\sum _k p_k|\psi ^{(k)}_{AB}\rangle \langle \psi ^{(k)}_{AB}|\)  , [2]}. By straightforward derivation we find that
\(\widehat{V}(|\psi _{AB}\rangle )=\frac{1}{2}\mathrm {E}^2(|\psi _{AB}\rangle )\) 
and \(\widehat{V}_c(\rho _{AB})=\frac{1}{2}\min _{\lbrace p_k,|\psi ^{(k)}_{AB}\rangle \rbrace } \sum _k p_k\mathrm {E}^2(|\psi _{AB}^{(k)}\rangle )\) . Equivalently,
\(E(\rho _{AB})=\min _{\lbrace p_k, |\psi ^{(k)}_{AB}\rangle \rbrace } \sum _k p_k\sqrt{2\widehat{V}(|\psi _{AB}^{(k)}\rangle )}\) , namely, the
entanglement measure concurrence can be expressed in terms of the
variance.
